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On a wave equation with supercritical interior and boundary sources and damping terms. (English) Zbl 1244.35092
In this article the following system is investigated: $$ \cases u_{tt}-\Delta u+g_0(u_t)=|u|^{p-1}u \quad\text{in } \Omega\times [0, \infty),\\ \partial_{\nu} u+u+g(u_t)=|u|^{k-1}u \quad\text{on } \Gamma\times [0, \infty),\\ u(0)=u_0\in H^1(\Omega),\ u_t(0)=u_1\in L^2(\Omega), \endcases \tag1 $$ where $\Omega\subset \Bbb R^3$ is a bounded open set with sufficiently smooth boundary $\Gamma$, the maps $g_0(s)$ and $g(s)$ are monotone and represent the interior and boundary dissipation, the Nemytski operators $|u|^{p-1}u$, $p>1$, $|u|^{k-1}u$, $k>1$, model the interior and boundary sources respectively. The authors give conditions for $g_0$, $g$ and $p$, $k$ under which the problem (1) has unique global weak solution $u\in {\cal C}([0, \infty), H^1(\Omega))$. The exponential and algebraic uniform decay rates of the finite energy are established. The authors prove a blow up result for weak solutions of (1) with nonnegative initial energy.

35L71Semilinear second-order hyperbolic equations
35A01Existence problems for PDE: global existence, local existence, non-existence
35B35Stability of solutions of PDE
35L20Second order hyperbolic equations, boundary value problems
35B33Critical exponents (PDE)
35B40Asymptotic behavior of solutions of PDE
35B44Blow-up (PDE)
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